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GLASNIK MATEMATIČKI Vol. 44(64)(2009), BIMORPHISMS OF A pro -CATEGORY Nikola Koceić Bilan University of Split, Croatia Abstract. Every morphism of an abstract coarse shape category Sh (C,D) can

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GLASNIK MATEMATIČKI Vol. 44(64)(2009), BIMORPHISMS OF A pro -CATEGORY Nikola Koceić Bilan University of Split, Croatia Abstract. Every morphism of an abstract coarse shape category Sh (C,D) can be viewed as a morphism of the category pro -D (defined on the class of inverse systems in D), where D is dense in C. Thus, the study of coarse shape isomorphisms reduces to the study of isomorphisms in the appropriate category pro -D. In this paper bimorphisms in a category pro -D are considered, for various categories D. We discuss in which cases pro -D is a balanced category (category in which every bimorphism is an isomorphism). We are interested in the question whether the fact that one of the categories: D, pro-d and pro -D is balanced implies that the other two categories are balanced. It is proved that if pro -D is balanced then D is balanced. Further, if D admits sums and products and pro -D is balanced then pro-d is balanced. In particular, pro -C is balanced for C = Set (the category of sets and functions) and C = Grp (the category of groups and homomorphisms). 1. Introduction The coarse shape theory was introduced and studied by N. Uglešić and the author in the joint paper [4]. In that paper, among others, the coarse shape category of topological spaces Sh Sh (HTop,HPol) has been constructed. The corresponding classification of topological spaces, induced by isomorphisms of Sh, is strictly coarser than the standard shape type classification. One can apply the same construction for any category pair (C, D), where D is dense in C (in the shape-theoretical sense [6]), to obtain an abstract coarse shape category Sh (C,D), having C-objects for the object class. The category Sh (C,D) is constructed from the category pro -D which 2000 Mathematics Subject Classification. 18A20, 55P55. Key words and phrases. Category, monomorphism, epimorphism, bimorphism, balanced category, pro-categories, topological space, polyhedron. 155 156 N. KOCEIĆ BILAN is defined on the class of inverse systems in D. For any pair X, Y of C- objects, every coarse shape morphism F Sh (C,D) (X, Y ) is represented by a unique morphism f pro -D (X, Y ) between inverse systems X and Y, i.e. Sh (C,D) (X, Y ) pro -D (X, Y ), where p : X X and q : Y Y are D-expansions of objects X and Y, respectively. Thus, the category Sh (C,D) is obtained via the category pro -D in the same manner as the abstract shape category Sh (C,D) is obtained via the category pro-d. Namely, the categories pro-d and pro -D have the same object class (inverse systems in D), but sets of morphisms are much larger in pro -D. Since a certain faithful functor J J D : pro-d pro -D, keeping the objects fixed, has been constructed, one may consider pro-d to be a subcategory of pro -D. Therefore, we may write (1) D pro-d pro -D. In any category the problem of detecting isomorphisms is essential. It can be readily seen that the discussion about coarse shape isomorphisms (isomorphisms in Sh (C;D)) reduces to the studying of isomorphisms in the category pro -D. For many familiar categories (Set - the category of sets and functions, Grp - the category of groups and homomorphisms, Cpt - the category of compact Hausdorff spaces) every morphism which is simultaneously a monomorphism and an epimorphism (called bimorphism) is an isomorphism. Such a category is called balanced. Notice that if a category is balanced, then, generally, its subcategory (or supercategory) needs not to be balanced. In this paper we are interested whether the fact that one of the categories in (1) is balanced implies that the other categories in (1) are balanced. 2. Preliminaries Let us recall the basic facts about pro-categories (see [6]) as well as pro - categories (see [4]). Let C be a category. An inverse system in C, denoted by X = (X λ, p λλ, Λ), consists of a directed preordered set (Λ, ), of C-objects X λ for each λ Λ, and of C-morphisms p λλ : X λ X λ (p λλ = 1 Xλ ), for each related pair λ λ in Λ, such that p λλ p λ λ = p λλ, whenever λ λ λ. A morphism of inverse systems (f, f µ ) : X Y = (Y µ, q µµ, M) consists of a function f : M Λ, and of C-morphisms f µ : X f(µ) Y µ for each µ M, such that, for every related pair µ µ, there exists a λ Λ, λ f(µ), f(µ ), such that f µ p f(µ)λ = q µµ f µ p f(µ )λ. Let X and Y be two inverse systems over the same index set Λ. A morphism (1 Λ, f λ ) : X Y is called a level morphism of inverse systems, provided BIMORPHISMS OF A pro -CATEGORY 157 f λ p λλ = q λλ f λ, for every related pair λ λ. A morphism (f, f µ ) : X Y is said to be equivalent to a morphism (f, f µ) : X Y, denoted by (f, f µ ) (f, f µ), provided each µ M admits a λ Λ, λ f(µ), f (µ), such that f µ p f(µ)λ = f µ p f (µ)λ. The equivalence class [(f, f µ )] of an (f, f µ ) is denoted by f. The composition of equivalence classes is well defined by putting gf = [(g, g ν )][(f, f µ )] = [(g, g ν )(f, f µ )]. The corresponding quotient category having all inverse systems X in C for objects, all equivalence classes f of morphisms of inverse systems for morphisms and 1 X = [(1 Λ, 1 Xλ )] for the identity morphism on an X is denoted by pro-c and is called the pro-category for the category C. We may treat each C-morphism f : X Y as a morphism in pro-c by putting f = (f) : (X) (Y ), where (X) and (Y ) are the rudimentary inverse systems. A morphism f is said to be induced by f. In this way, a category C can be considered as a subcategory of pro-c. An S -morphism of inverse systems, (f, fµ) n : X Y, consists of a function f : M Λ, called the index function, and of a set of C-morphisms fµ n : X f(µ) Y µ, n N, µ M, such that, for every related pair µ µ in M, there exists a λ Λ, λ f(µ), f (µ ), and there exists an n N so that, for every n n, fµ n p f(µ)λ = q µµ fµ n p f(µ )λ. If the index function f is increasing and, for every pair µ µ, one may put λ = f(µ ), then (f, fµ n ) is said to be a simple S -morphism. If, in addition, M = Λ and f = 1 Λ, then (1 Λ, fλ n) is said to be a level S -morphism. The composition of S -morphisms of inverse systems is defined as follows: If (f, fµ) n : X Y and (g, gν) n : Y Z, then (g, gν)(f, n fµ n ) = (h, h n ν) : X Z, where h = fg and h n ν = gνf n g(ν) n. The identity S -morphism on X is an S -morphism (1 Λ, 1 n X λ ) : X X, consisting of the identity function 1 Λ and of the identity morphisms 1 n X λ = 1 Xλ in C, for every n N and every λ Λ. An S -morphism (f, fµ n ) : X Y of inverse systems in C is said to be equivalent to an S -morphism (f, f µ n ) : X Y, denoted by (f, fµ) n (f, f µ n), provided every µ M admits a λ Λ, λ f(µ), f (µ), and an n N, such that, for every n n, f n µ p f(µ)λ = f n µ p f (µ)λ. The relation is an equivalence relation among S -morphisms of inverse systems in C. The equivalence class [(f, f n µ)] of an S -morphism (f, f n µ) : X Y is briefly denoted by f. 158 N. KOCEIĆ BILAN The category pro -C has as objects all inverse systems X in C and as morphisms all equivalence classes f = [(f, f n µ)] of S -morphisms (f, f n µ). Since the equivalence relation respects the composition of S -morphisms, a composition in pro -C is well defined by putting g f = h [(h, h n ν )], where (h, h n ν) = (g, gν n )(f, fµ) n = (fg, gνf n g(ν) n ). For every inverse system X in C, the identity morphism in pro -C is 1 X = [(1 Λ, 1 Xλ )]. A functor J J C : pro-c pro -C is defined as follows. It keeps objects fixed, i.e. J (X) = X, for every inverse system X in C. If f pro-c(x, Y ) and if (f, f µ ) is any representative of f, then a morphism J (f) = f = [(f, fµ n )] pro -C(X, Y ) is represented by an S -morphism (f, fµ n ) where fµ n = f µ for all µ M and n N. The morphism f is said to be induced by f. Since the functor J is faithful, we may consider the category pro-c as a subcategory of pro -C. Thus, every morphism f in pro-c can also be considered as a morphism of the category pro -C. Recall that an index set is said to be cofinite if its preordering is an ordering and every µ M has finitely many predecessors. Concerning inverse systems indexed by a cofinite index set, we have a very useful lemma which easily follows from [4, Lemma 10]. Lemma 2.1. Let X = (X λ, p λλ, Λ) and Y = (Y µ, q µµ, M) be inverse systems in C with M cofinite. Then every morphism f : X Y of pro -C admits a simple representative (f, fµ n ) : X Y. Moreover, if (f, f µ n ) is any simple representative of f, then, for every µ M, there exists n µ N such that, for every µ µ and every n n µ, f n µ p f (µ )f (µ) = q µ µf n µ In general, a morphism f : X Y in pro -C does not admit a level representative. However, the following reindexing theorem will help to overcome some technical difficulties concerning this fact. Theorem 2.2. Let f pro -C(X, Y ). Then there exist inverse systems X and Y in C having the same cofinite index set (N, ), there exists a morphism f : X Y having a level representative (1 N, f ν ) and there exist isomorphisms i : X X and j : Y Y in pro -C, such that the following diagram in pro -C commutes: f X Y i j. X f Y The analogous theorem ([6, Theorem ]) holds in every pro-category pro-c. Concerning many problems, these reindexing theorems allow to assume that each morphism in pro-c or pro -C admits a level representative. BIMORPHISMS OF A pro -CATEGORY 159 Moreover, we can assume that both inverse systems are indexed by the same cofinite index set. 3. Isomorphisms in a pro -category In this section we are dealing with isomorphisms in pro -C which are induced by morphisms in pro-c. Notice that, if B is a subcategory of A, and a morphism f B (X, Y ) A(X, Y ) is an isomorphism in a category B then f is an isomorphisms in a category A, as well. But the converse is not generally true. Since pro-c can be viewed as a subcategory of pro -C the following question naturally arises: Problem 3.1. If a morphism f = J (f) pro (X, Y ), induced by f pro-c (X, Y ), is an isomorphism of pro -C, is it true that the morphism f is an isomorphism of pro-c? Before we answer the above question affirmatively (see Theorem 3.2, below) let us recall an analogue of the well known Morita lemma ([6, Theorem 2.2.5]) which characterizes isomorphisms in a pro -category ([4, Theorem 5]). Theorem 3.1. Let X = (X λ, p λλ, Λ) and Y = (Y λ, q λλ, Λ) be inverse systems in C over the same index set. Let a morphism f : X Y in pro -C admit a level representative (1 Λ, fλ n). Then f is an isomorphism if and only if, for every λ Λ, there exist a λ λ and an n N such that, for every n n, there exists a morphism h n λ : Y λ X λ in C, such that the following diagram in C commutes: (2) X λ X λ fλ n hn λ տ fn λ Y λ Y λ Theorem 3.2. A morphism f : X Y in pro-c is an isomorphism if and only if the induced morphism f = J (f) : X Y is an isomorphism of pro -C. Proof. Since J is a functor, the necessity holds trivially. Conversely, suppose that the induced morphism f = J (f) is an isomorphism of pro -C. By the reindexing theorem ([6, Theorem 1.1.3]) there is no loss of generality in assuming that f is represented by a level morphism (1 Λ, f λ ). Hence, the induced morphism f = J (f) : X Y in pro -C is represented by the induced level S -morphism (1 Λ, fλ n), fn λ = f λ, for all n N, λ Λ. Since, f is an isomorphism, the level representative (1 Λ, fλ n ) satisfies the condition of Theorem 3.1. That means that, for every λ Λ, there exist a λ λ and an n N such that, for every n n, there exists a morphism h n λ : Y λ X λ in C, so that diagram (2) in C commutes. Now, for every n n, it follows that f λ h n λ = f n λ h n λ = q λλ. 160 N. KOCEIĆ BILAN and h n λ f λ = h n λ fλ n = p λλ, which means that the morphism (1 Λ, f λ ) fulfills the condition of the Morita lemma in pro-c ([6, Theorem 2.2.5]). Therefore, f = [(1 Λ, f λ )] is an isomorphism of pro-c. Notice that, by Morita lemma, it follows that a morphism f : X Y is an isomorphism of a category C if and only if the induced morphism (f) : (X) (Y ) in pro-c is an isomorphism. Therefore, by Theorem 3.2, the following corollary holds Corollary 3.3. Let f : X Y be a morphism of a category C. Then the following three conditions are equivalent: (i) f : X Y is an isomorphism of C. (ii) The induced morphism (f) : (X) (Y ) is an isomorphism of pro-c. (iii) The induced morphism f = J ((f)) : (X) (Y ) is an isomorphism of pro -C. 4. Bimorphisms in a pro -category We are interested in determining under what conditions the fact that one of the categories in (1) is balanced implies that the two other categories in (1) are balanced. Theorem 4.1. If pro -C or pro-c is a balanced category, then C is also a balanced category. Proof. Assuming on the contrary, i.e. if C is not a balanced category, then there exists a bimorphism f : X Y of C which is not an isomorphism. According to [5, Corollary 3 and Corollary 6], every bimorphism of C is also a bimorphism of pro-c and pro -C. Thus, the induced morphisms (f) in pro-c and J ((f)) in pro -C are bimorphisms. This bimorphisms cannot be isomorphisms of pro-c nor pro -C, respectively, because, by Corollary 3.3, it would imply that f is a C-isomorphism. Hence, pro-c and pro -C are not balanced, which is a contradiction. Theorem 4.2. Let C be a category admitting sums and products. If pro - C is balanced, then pro-c is also a balanced category. Proof. Assuming on the contrary, i.e. if pro-c is not a balanced category, then there exists a bimorphism f : X Y of pro-c which is not an isomorphism. According to [5, Theorem 3 and Theorem 5], every bimorphism of pro-c is also a bimorphism of pro -C. Thus, the induced morphism f = J (f) is a bimorphism of pro -C. Now, f cannot be an isomorphism, because, by Theorem 3.2, that would imply that f is an isomorphism. That means, the morphism f is a bimorphism, but not an isomorphism of pro -C. Hence, pro -C is not balanced, which is a contradiction. BIMORPHISMS OF A pro -CATEGORY 161 The question: Is the category pro-c balanced when C is balanced? posed in [1] was recently answered negatively in [2]. It has been proved ([2], Proposition 3.12.) that the category whose objects are compact connected spaces and morphisms are maps is a balanced category, but the corresponding procategory is not balanced. Since the category of compact connected spaces is balanced but it does not admit sums, we cannot apply Theorem 4.2. Consequently, the following problem remains open. Problem 4.1. If C be a balanced category, is the category pro -C balanced? We will prove that in two important cases of balanced categories, C = Set and C = Grp, the category pro -C is balanced. Definition 4.3. We say that a category C -additive if it has zero-objects, admits sums and products, every morphism of C has the kernel and cokernel and every morphism set C (X, Y ) is a group such that the composition : C (X, Y ) C (Y, Z) C (X, Z) is bilinear. A zero-object of a category C is denoted by 0. For any two objects, a unique zero-morphism which factorizes through 0 is denoted by o XY : X Y (briefly o). Notice that a zero-morphism is the unit element of the group C (X, Y ) (see [3]), therefore we denote the group operation additively, although C (X, Y ) is not, in general, an abelian group. The kernel of a morphism f : X Y we denote by kerf : N X, while cokerf : Y C denotes the cokernel of f. Proposition 4.4. Let C be a -additive category. Let X = (X λ, p λλ, Λ) and Y = (Y λ, q λλ, Λ) be inverse systems in C over the same cofinite index set. Let a morphism f : X Y in pro -C admit a level representative (1 Λ, fλ n). A morphism f is a monomorphism if and only if (1 Λ, fλ n) satisfies the following condition: (M-Add) For every λ Λ, there exist a λ λ and an n 0 N such that, for every n n 0, p λλ (kerfλ n ) = o. Further, f is an epimorphism if and only if (1 Λ, fλ n ) satisfies the following condition: (E-Add) For every λ Λ, there exists a λ λ and an n 0 N such that, for every n n 0, (cokerfλ n)q λλ = o. Proof. Assume that f is a monomorphism. Then, according to [5, Theorem 4], a level representative (1 Λ, fλ n ) satisfies condition (M) of the same theorem. For an arbitrary λ, let λ λ be as in condition (M). Let us put u n = kerfλ n : Nn λ X λ and vn : Nλ n X λ, vn = o, for every n N. Obviously, fλ n un = fλ n vn = o holds for every n N. Now by (M), there exist an n 0 N such that, p λλ kerfλ n = p λλ un = p λλ v n = o, for every n n 0, which establishes (M-Add). Conversely, suppose that (1 Λ, fλ n) 162 N. KOCEIĆ BILAN satisfies (M-Add). Then, for an arbitrary λ Λ, there exist a λ λ and an n 0 N as in condition (M-Add). Let (u n ), (v n ) be a pair of sequences of C-morphism u n, v n : P n X λ, P n Ob (C), such that fλ n un = fλ n vn, for every n N. It implies fλ n (un v n ) = o, for every n N. Consider the kernel kerfλ n : Nn λ X λ of fn λ, for every n N. According to the universal property of a kernel, for every n N, there exists a morphism h n : P n Nλ n such that (kerfn λ )hn = (u n v n ). Now, by(m-add), it follows that p λλ (u n v n ) = p λλ (kerfλ n )hn = o, for every n n 0, which implies that p λλ u n = p λλ v n, for every n n 0. Thus, (1 Λ, fλ n ) satisfies condition (M) of [5, Theorem 4], and, by the same theorem, f is a monomorphism. Analogously, one can prove a statement concerning an epimorphism, i.e. the equivalence, for a -additive category, between (E-Add) and the condition (E) of [5, Theorem 2]. Example 4.5. Clearly, the category Grp is a -additive category. Thus, we can apply Proposition 4.4 to characterize epimorphisms and monomorphisms in pro -Grp. We point out that a group operation is denoted additively and the unit element is denoted by 0, for every group, not necessarily abelian. Recall that a kernel of a homomorphism f : X Y is the inclusion i : f 1 (0) X (the group f 1 (0) is usually denoted by kerf) and a cokernel of f is the quotient homomorphism p : Y Y/ Im f. One can easily verify that, for C = Grp, (E-Add) is equivalent to the following condition: (E-Grp) For every λ Λ, there exist a λ Λ, λ λ, and an n 0 N such that Im q λλ Im f n λ, for all n n 0. Further, for C = Grp, (M-Add) is equivalent to the condition (M-Grp) For every λ Λ there exist a λ Λ, λ λ, and an n 0 N such that kerf n λ kerp λλ, for every n n 0. Remark 4.6. Let us consider a morphism f = [(f, f µ )] in pro-grp. For the induced morphism f = [( f, f n µ)] = J (f), f n µ = f µ, for every n N, Proposition 4.4 allows to put n 0 = 1, for every λ. Thus, for the induced morphism, the dependence on the indices n in the conditions(e-grp) and(m-grp) vanishes. Consequently, the conditions (E-Grp) and (M-Grp) in the subcategory pro-grp become the well-known conditions for f being a monomorphism and an epimorphism of pro-grp respectively (see [6, Theorem and Theorem 2.2.3]). Theorem 4.7. The categories pro -Set and pro -Grp are balanced. Proof. We need to prove that every bimorphism in pro -Grp (pro - Set) is an isomorphism. Suppose f : X Y is a bimorphism of pro - Grp (pro -Set). By the reindexing theorem (Theorem 2.2), there is no loss of generality in assuming that f is represented by a level morphism (1 Λ, f λ ) : X Y where X = (X λ, p λλ, Λ) and Y = (Y λ, q λλ, Λ) are inverse BIMORPHISMS OF A pro -CATEGORY 163 systems in pro -Grp (pro -Set) over the same cofinite index set. Therefore, by Lemma 2.1, for every λ Λ, there exist an n λ N such that, for every λ λ and every n n λ, f n λ p λ λ = q λ λf n λ. First, let us show that a bimorphism f of pro -Grp is an isomorphism. By Example 4.5, (1 Λ, f λ ) fulfills conditions (M-Grp) and (E-Grp). Hence, for an arbitrary λ Λ, there exist a λ λ and an n 0 N such that (3) kerp λλ kerf n λ, for every n n 0. Further, for this λ, there exist a λ λ and an n 0 such that (4) Im f n λ Im q λ λ, n n 0. Let u

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